Non-linear viscoelastic models

Viscoelasticity has already been introduced in Chapter 1, based on linear viscoelasticity. However, in polymer processing large deformations are imposed on the material, requiring the use of non-linear viscoelastic models. There are two types of general non-linear viscoelastic flow models the differential type and the integral type. 

D. Aciemo, F. P. La Mantia, G. Marrucci, and G. Titomanlio, A Non-linear Viscoelastic Model with Structure-dependent Relaxation Times I. Basic Formulation, J. Non-Newt. Fluid Meek, 1, 125-146 (1976). 

The non linear viscoelasticity of various particles filled rubber is addressed in range of studies. It is found that the carbon black filled-elastomer exhibit quasi-static and dynamic response of nonlinearity. Hartmann reported a state of stress which is the superposition of a time independent, long-term, response (hyperelastic) and a time dependent, short-term, response in carbon black filled-rubber when loaded with time-dependent external forces. The short term stresses were larger than the long term hyperelastic ones. The authors had done a comparative study for the non linear viscoelastic models undergoing relaxation, creep and hysteresis tests [20-22]. For reproducible and accurate viscoelastic parameters an experimental procedure is developed using an ad hoc nonlinear optimization algorithm. 

The discussion so far has been dominated by one-dimensional behaviour, reflecting the most convenient and customary materials testing methods. However, any engineering application will be for a three-dimensional body, subject to multi-axial stresses. It is now feasible to implement non-linear viscoelastic models in numerical schemes to perform analyses of structures, and this is often the motivation for generalising a viscoelastic theory to two or three dimensions. 

Jindrieh D, Zhou Y, Becker Theodore (2003) Non-linear viscoelastic models predict finger tip pulp force-displacement characteristics during voluntary tapping. Journal of Biomechanics 36 497-503... 

Because of the assumption that linear relations exist between shear stress and shear rate (equation 3.4) and between distortion and stress (equation 3.128), both of these models, namely the Maxwell and Voigt models, and all other such models involving combinations of springs and dashpots, are restricted to small strains and small strain rates. Accordingly, the equations describing these models are known as line viscoelastic equations. Several theoretical and semi-theoretical approaches are available to account for non-linear viscoelastic effects, and reference should be made to specialist works 14-16 for further details. 

The term y(t,t ) is the shear strain at time t relative to the strain at time t. The use of a memory function has been adopted in polymer modelling. For example this approach is used by Doi and Edwards11 to describe linear responses of solution polymers which they extended to non-linear viscoelastic responses in both shear and extension. 

The most surprising result is that such simple non-linear relaxation behaviour can give rise to such complex behaviour of the stress with time. In Figure 6.3(b) there is a peak termed a stress overshoot . This illustrates that materials following very simple rules can show very complex behaviour. The sample modelled here, it could be argued, can show both thixotropic and anti-thixotropic behaviour. One of the most frequently made non-linear viscoelastic measurements is the thixotropic loop. This involves increasing the shear rate linearly with time to a given... 

Contents Chain Configuration in Amorphous Polymer Systems. Material Properties of Viscoelastic Liquids. Molecular Models in Polymer Rheology. Experimental Results on Linear Viscoelastic Behavior. Molecular Entan-lement Theories of Linear iscoelastic Behavior. Entanglement in Cross-linked Systems. Non-linear Viscoelastic-Properties. 

The effect of gas compression on the uniaxial compression stress-strain curve of closed-cell polymer foams was analysed. The elastic contribution of cell faces to the compressive stress-strain curve is predicted quantitatively, and the effect on the initial Young s modulus is said to be large. The polymer contribution was analysed using a tetrakaidecahedral cell model. It is demonstrated that the cell faces contribute linearly to the Young s modulus, but compressive yielding involves non-linear viscoelastic deformation. 3 refs. 

The function F(t — t ) is related, as with the temporary network model of Green and Tobolsky (48) discussed earlier, to the survival probability of a tube segment for a time interval (f — t ) of the strain history (58,59). Finally, this Doi-Edwards model (Eq. 3.4-5) is for monodispersed polymers, and is capable of moderate predictive success in the non linear viscoelastic range. However, it is not capable of predicting strain hardening in elongational flows (Figs. 3.6 and 3.7). 

On a global scale, the linear viscoelastic behavior of the polymer chains in the nanocomposites, as detected by conventional rheometry, is dramatically altered when the chains are tethered to the surface of the silicate or are in close proximity to the silicate layers as in intercalated nanocomposites. Some of these systems show close analogies to other intrinsically anisotropic materials such as block copolymers and smectic liquid crystalline polymers and provide model systems to understand the dynamics of polymer brushes. Finally, the polymer melt-brushes exhibit intriguing non-linear viscoelastic behavior, which shows strainhardening with a characteric critical strain amplitude that is only a function of the interlayer distance. These results provide complementary information to that obtained for solution brushes using the SFA, and are attributed to chain stretching associated with the space-filling requirements of a melt brush. 

This assumption of a linear relationship between stress and strain appears to be good for small loads and deformations and allows for the formulation of linear viscoelastic models. There are also non-linear models, but that is an advanced topic that we won t discuss. There are two approaches we can take here. The first is to develop simple mathematical models that are capable of describing the structure of the data (so-called phenomenological models). We will spend some time on these as they provide considerable insight into viscoelastic behavior. Then there are physical theories that attempt to start with simple assumptions concerning the molecules and their interactions and... 

However, although it has some thermodynamic consistency, the latter model failsto describe the non linear viscoelastic behaviour properties, especially in shear, wherein the shear-thinning behaviour of the viscosity and of the normal stress coefficients are not predicted. As a consequence, more complex... 

The correlation between rheology and thermodynamics is likely to prove a fruitful area for investigation in the future. Very little is as yet known about the detailed mechanisms of non-linear viscoelastic flows, such as those involved in large-amplitude oscillatory shear. Mesoscopic modelling will no doubt throw light on the role of defects in such flows. This is likely to involve both analytical models, and mesoscopic simulation techniques such as Lattice... 

Materials can show linear and nonlinear viscoelastic behavior. If the response of the sample (e.g., shear strain rate) is proportional to the strength of the defined signal (e.g., shear stress), i.e., if the superposition principle applies, then the measurements were undertaken in the linear viscoelastic range. For example, the increase in shear stress by a factor of two will double the shear strain rate. All differential equations (for example, Eq. (13)) are linear. The constants in these equations, such as viscosity or modulus of rigidity, will not change when the experimental parameters are varied. As a consequence, the range in which the experimental variables can be modified is usually quite small. It is important that the experimenter checks that the test variables indeed lie in the linear viscoelastic region. If this is achieved, the quality control of materials on the basis of viscoelastic properties is much more reproducible than the use of simple viscosity measurements. Non-linear viscoelasticity experiments are more difficult to model and hence rarely used compared to linear viscoelasticity models. 

More recently, a new, viscoelastic-plastic model for suspension of small particles in polymer melts was proposed [Sobhanie et al., 1997]. The basic assumption is that the total stress is divided into that in the matrix and immersed in it network of interacting particles. Consequently, the model leads to non-linear viscoelastic relations with yield function. The latter is defined in terms of structure rupture and restoration. Derived steady state and dynamic functions were compared with the experimental data. 

Dus, S. J., and Kokini, J. L. (1990). Prediction of the non-linear viscoelastic properties of a hard wheat flour dough using the Bird-Carreau constitutive model. J. Rheol. 34(7), 1069-1084. 

The models will assume linearity, as expressed in equations (7.1) and (7.2). These equations apply only at small strains and considerations in this chapter are restricted to such small strains. It should, however, be noted that real polymers are often non-linear even at small strains. The subject of non-linear viscoelasticity lies outside the scope of this book but is considered in the books referred to as (1) in section 7.7. 

Caruthers, J. M., Adolf, D. B., Chambers, R. S., and Shrikhande, P, A thermodynamically consistent, non-linear viscoelastic approach for modelling glassy polymers. Polymer, 45, 4577-4597 (2004). 

K-BKZ model for non-linear viscoelastic body proposed by Kaye... 

Several investigators in the field of rheology have suggested that free volume is a good unifying parameter to describe changes in the timescale of material response in polymers. Free volume is the portion of the specific volume of the material that is unoccupied by the molecules. Researchers have applied the concept of free volume to develop a non-linear viscoelastic constitutive relationship [2] as well as for modeling coupled diffusion in viscoelastic materials [1]. 

The models have been developed mainly for semi-crystalline polymers, which in general show the largest mechanical anisotropy, but some of the discussion is equally relevant to oriented non-crystalline polymers. Although an oriented polymer is strictly a non-linear viscoelastic solid (see Chapters 10 and 11) the present discussion is restricted to theoretical models which represent linear elastic or linear viscoelastic behaviour. 

Entanglements of flexible polymer chains contribute to non-linear viscoelastic response. Motions hindered by entanglements are a contributor to dielectric and diffusion properties since they constrain chain dynamics. Macromolecular dynamics are theoretically described by the reptation model. Reptation includes fluctuations in chain contour length, entanglement release, tube dilation, and retraction of side chains as the molecules translate using segmental motions, through a theoretical tube. The reptation model shows favourable comparison with experimental data from viscoelastic and dielectric measurements. The model reveals much about chain dynamics, relaxation times and molecular structures of individual macromolecules. 

A phenomenological model has been proposed for the non-linear viscoelastic behaviour of thermorheologjcally complex polymer glasses prior to and including yield. The approach was based upon stress additivity. A linear viscoelastic material will exhibit stress-strain additivity. The molecular processes modelled were resolved into two parallel processes, each with a characteristic relaxation time spectrum. The model described the yield behaviour and creep experiments at increasing stress. " ... 

Starch nanocrystals were used to reinforce a non-vulcanised NR matrix. The NR was not vulcanised to enhance biodegradability of the total biocomposite. Non-linear dynamic mechanical experiments demonstrated a strong reinforcement by starch nanocrystals, with the presence of Mullins and Payne effects. The Payne effect was able to be predicted using a filler-filler model (Kraus model) and a matrix-filler model (Maier and Goritz model). The Maier and Goritz model showed that adsorption-desorption of NR onto the starch surface contributed the non-linear viscoelasticity. The Kraus model confirmed presence of a percolation network. ... 

Non-linear viscoelastic mechanical behaviour of a crosslinked sealant was interpreted as due to a Mullins effect. The Mullins effect was observed for a series of sealants under tensile and compression tests. The Mullins effect was partially removed after a mechanical test, when a long relaxation time was allowed, that is the modulus increased over time. Non-linear stress relaxation was observed for pre-strained filler sealants. Time-strain superposition was used to derive a model for the filled sealants. Relaxation over long periods demonstrates that the Mullins effect is caused by non-equilibrium with experimental conditions being faster than return to the initial state. If experiments were conducted over times of the order of a day there may be no Mullins effect. If a filled elastomer were only required to perform its function once per day then each response might be linear viscoelastic. 

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